FIGURE 2. Einstein’s schematic 1916 illustration of the thought experiment about lightning bolts hitting the front and back of a train. M’ is the observer sitting on the train, M is the woman at the station, and A and B represent the points where the lightning bolts hit the ground. In this rendition, the velocity v represents the train speeding to the right at a constant rate. It is, however, just as legitimate—indeed, essential for the scenario—that M’ perceives himself as stationary and M and her embankment as zipping off to the left at a constant velocity of –v. The speed of light, c, which is constant for all observers, is not depicted. Source: Albert Einstein, Relativity: The Special and the General Theory, tr. Robert W. Lawson (New York: Three Rivers Press, 1961 [1916]), 29.
Here’s how to resolve the difference of perception. From the frame of reference of the station, the two lightning bolts are simultaneous, and the reason that the man sees them staggered is that he is rushing toward the light beam heading from the front of the train, encountering it first, and the light beam from the rear takes some extra instants to catch up to him. But—and this is the crucial point—since the motion here is constant, it is just as correct for the man on the train to believe that he is stationary and the platform is moving in the opposite direction away from him at Star Trek speed. In his frame of reference, the lightning bolt at the front of the train happened first, the one at the rear second, but the platform was rushing toward the rear of the train so fast that the woman measured the beam from the back just as the one from the front finally reached her. The notion of simultaneity depends on one’s frame of reference. From this basic qualitative scenario, the seemingly paradoxical properties of which stem entirely from the assumptions that the frames of reference are equivalent and that the speed of light moves at a constant velocity in every frame, one can derive the astonishing consequences of special relativity.
If one is accelerating, though, things work differently. Imagine now that you are riding in a windowless elevator being accelerated upward. You feel your weight on the floor quite firmly as it pushes against your feet. If a light beam now were to penetrate the side of the elevator, it would seem to follow an arc downward, as the accelerating elevator zoomed upward. That’s odd: light follows a curved path. But here’s the wrinkle Einstein noticed: There is no way to tell from within the elevator whether you are being accelerated upward by an incredibly powerful winch attached to a cable on the roof or whether you are simply standing at rest in a gravitational field. In both situations, the light’s path would bend, and you would feel pressure from the floor. If one followed the consequences out to their conclusion, this would mean that any attempt to generalize special relativity to accelerated frames would be indistinguishable from a theory of gravitation. This was Einstein’s speculation in 1907 in his article for the Jahrbuch.
This insight relied on an assumption that Einstein understood to be universal, what he called the “equivalence principle.” In classical physics, there were two different understandings of mass, our measure of the quantity of matter in any given body: gravitational mass, or the quantity of matter that exerted a gravitational force according to Newton’s law of universal gravitation; and inertial mass, or the quantity of matter that resisted changes in motion, as in Newton’s second law that force equals mass times acceleration (F = ma). The explanation for motion was entirely different depending on whether one had inertial or gravitational mass in mind; the odd thing is that the physical results were always the same. As Einstein noted in 1907, “This proportionality between inertial and gravitational mass holds however without exception for all bodies to the level of precision heretofore attained, so that we must until there is justification of the contrary consider it generally applicable.”26 If the equivalence principle always held true, as the detailed measurements by Hungarian physicist Roland von Eötvös indicated they did, then perhaps there was only one kind of mass, and one kind of physics.27 A generalization of special relativity would necessarily be a theory of gravity.
It was an audacious proposal. “I am extremely excited about your new conception of gravitation,” physicist Max von Laue wrote to Einstein upon reading the Jahrbuch essay.28 Einstein was clearly itching to see whether his hunch would bear out. The following year he wrote to Arnold Sommerfeld in Munich to underscore the potential of the idea: “The fact that in a gravitational field all bodies are accelerated equally inclines one precisely very much to the idea that an accelerated coordinate system and an acceleration-free coordinate system with a homogeneous gravitational field appear as completely equivalent things. One arrives on the grounds of this assumption at very plausible consequences.”29 Not everyone was equally enthusiastic. In 1911, in the first major book on relativity theory, von Laue thought that the probability that Einstein’s intuition would be validated was relatively low. He ended his summary of the consequences of the equivalence principle with a series of questions about the causes of gravity, the results of gravitational momentum, and its effect on light beams. “No one can presently answer all this,” he concluded, “and precisely the circumstance that Newton’s law, victorious so far, fully satisfies the astronomers, places the hope for an answer in the foreseeable future to a minimum.”30
Despite some initial notice taken of Einstein’s speculations at the end of his 1907 paper, by the time the physicist arrived in Prague in spring 1911, most established theoretical physicists would have agreed with von Laue’s pessimistic verdict: it was a nice notion, but not a likely one. This was not so much instinctive conservatism—among the architects of quantum theory, conservatism in physical speculation was not especially pronounced—as an assessment of opportunity costs and an induction about the unillustrious examples presented by past attempts to reform Newtonian gravity. The costs were straightforward enough: although there were, to be sure, some anomalies presented by Newtonian universal gravitation (such as the well-known mismatch between the theory and the observed advance of the perihelion of Mercury), surely the challenges in statistical mechanics and atomic physics were more pressing and more likely to yield empirical verification. As for the gallery of discarded explanations for the cause of gravity, there was little reason for optimism.31 Even a relativistic theory of gravity developed around 1911 by as gifted a theorist as Henri Poincaré, who had formulated his own special theory of relativity alongside Einstein’s, was largely ignored, given that it was not rooted in the dominant tradition of field theory.32
This climate of aloofness helps explain what has sometimes been posited as Einstein’s “delay” in exploiting his 1907 intuitions about the generalization of relativity. Looking back in 1949 on the four years between the Jahrbuch essay and his arrival at the Institute of Theoretical Physics on the Weinberggasse in Prague, Einstein had this to say: “Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that co-ordinates must have an immediate metrical meaning.”33 Notice the three additional years Einstein added to this account: he was not measuring the distance between his early notes and Prague, but between them and the formulation of full-blown general relativity in Berlin. Einstein had not only forgotten the distractions of quantum theory in postponing his work on general relativity, but reduced everything to the technical challenges that beset him after he began his collaboration with Grossmann in Zurich. Almost four decades later, Einstein had even forgotten that Prague had provided him the setting and the respite to devote himself to this question. He had also forgotten the static theory that he had left in tatters on his office floor at the German University. It is time to bring them back to light.
* * *
Light was at the core of the static theory, in two senses: the constancy of its speed and its potential to bend in a gravitational field. The speed of light dominated special relativity, making the novel conception of spacetime thinkable. Indeed, one of the first mentions of Einstein’s new turn toward full-time research on gene
ral relativity after his move to Prague placed particular emphasis on the issue of light’s speed. “The relativistic treatment of gravitation creates serious difficulties,” he wrote to Jakob Laub. “I consider it probable that the principle of the constancy of the velocity of light in its customary formulation only holds for spaces of constant gravitational potential.”34 Einstein’s willingness to relax his former insistence on the constancy of the speed of light in all frames of reference dominated all of his publications on the topic in Prague.35
This is especially noteworthy given the role that this stricture played in his 1905 work on the electrodynamics of moving bodies. The classic paper introducing special relativity begins with an elementary thought experiment concerning electromagnetic induction with a magnet and a coil of wire, and then immediately introduces two principles that ushered physics into a post-Newtonian age. First, Einstein expanded the classic Galilean principle of relativity, now insisting that the laws of physics—including those of electromagnetism—must be precisely the same in all inertial frames of reference, that is, all frames that are moving at a constant velocity with respect to each other. The goal of general relativity was also to incorporate accelerated frames of reference. (Think about being on an airplane: you know when you are taking off or landing because the acceleration thrusts you back into your seat or forward against your seatbelt; when you are flying at a constant velocity, however, as far as the physics of tossing a ball is concerned, you might as well be standing still.) The second principle is more counterintuitive: “Light in empty space always propagates with speed V, independent of the state of motion of the emitting body.”36 (Einstein would quickly adopt the convention of using c to label the speed of light.)
In the summer of 1911, Einstein had an idea that there was a connection between the speed of light and the gravitation potential. “I, by contrast, have on the basis of some rumination—already somewhat daring but that still has much going for it—arrived at the view,” he wrote to the Dutch physicist Willem Julius that August, “that the difference in gravitational potential could be a cause of the shifting of the [spectral] lines. From these considerations follows also a bending of light rays by gravitational fields.”37 We have already seen that the intuition that light rays would bend in a gravitational field followed from the thought experiment about the elevator hurtling through space. By June, Einstein had figured out how to calculate the degree of bending by modifying the gravitational potential, the factor that determined the strength of the gravitational field at each point of spacetime. As he then put it slightly more technically in the most influential of his Prague physics papers published in the Annalen der Physik on 1 September of that year:
We must for time measurement at a place that has a gravitational potential Φ relative to the origin use a clock that—set at the coordinate origin—runs (1 + Φ/c2) times slower than that clock with which time is measured at the origin. We call the speed of light at the origin point c0, so the speed of light c in a place with gravitational potential Φ is given through the equation
c = c0(1 + Φ/c2).
The principle of the constancy of the speed of light according to this theory does not hold in the same manner as it was preserved as an axiom in conventional relativity theory.38
Indeed it does not. The point of this equation is to argue that instead of using the classic Newtonian gravitational potential formula (known as the Poisson equation), one could instead track the changes of the gravitational potential Φ by introducing a speed of light that changes at every single point in spacetime. (It would be, however, constant for any two observers at the same spacetime point or experiencing the same gravitational potential.) One way of thinking about this is to return to Einstein’s classic equation of the relation of mass to energy, E = mc2. In a gravitational field, the energy would fluctuate, and he could capture that either by changing mass or changing the speed of light. He opted for the latter.
Why is this important? The title of this first Prague article on what would later be called the static theory, “On the Influence of Gravitation on the Propagation of Light,” makes the stakes fairly clear. “I have already sought an answer to the question of whether the propagation of light would be influenced through gravity in an article which appeared three years ago. I return to the topic because my presentation of the situation then did not satisfy me, all the more however because I now realize belatedly that one of the most important consequences of that observation is amenable to experimental testing,” he wrote in the opening paragraph. “That is, namely, that light beams which pass in the vicinity of the sun will according to the proposed theory experience a deflection by its gravitational field so that one would find a sensible enlargement of almost a second of arc of the angular distance of a fixed star appearing near the sun.”39 That would be large enough to be measured, but it would require a solar eclipse to be able to see starlight streaming past the sun. Jupiter, the most massive planet in our solar system, would have made for a more convenient observational test, but unfortunately he calculated that the effect would be 100 times weaker there than around the sun, too small for contemporary instruments to detect.40
If this paper had been all Einstein had published on the static theory, his Prague theory of gravitation would have gone down in history as a success. This prediction of the bending of starlight around massive bodies like the sun later became the most publicly ballyhooed empirical result of general relativity, and its renown transformed Einstein into the tremendously famous physicist we know today. That event, however, would not occur until spring 1919, when the British astronomer Arthur Stanley Eddington organized two expeditions to measure the potential bending of starlight during a total eclipse of the sun. Eddington’s own goals were not just physical: a devout Quaker, he was concerned with repairing the shattered relations among European nations due to the Great War, and a British test of a German pacifist’s theory of the universe seemed to fit the bill. The positive results of this test, touted in newspapers worldwide, were the impetus for Einstein’s ascension to international fame. Given the drama of Eddington’s adventure and its epic consequences, it is understandable that many think of the bending of starlight in the context of Anglo-German relations and then global notoriety, as Americans in California continued to probe the results (eventually confirming them), and reactionary anti-Einsteiners touted a “German” (i.e., not Jewish) precursor whom they argued should receive the credit. This focus on the transnational Einstein is very much valid.41 There is also, however, a complementary local story in Prague.
Einstein’s newfound interest in astronomical observations brought him into contact with Leo Wenzel Pollak, a demonstrator at the Institute for Cosmic Physics at the German University. Pollak found the claim of starlight bending around the sun to be intriguing, and he wrote about it to a colleague in Berlin named Erwin Finlay Freundlich in late August 1911—after Einstein had submitted his paper to the Annalen but before it had been published.42 Freundlich was excited by the idea, and he and Einstein entered into discussions about how it might be tested. Almost everything about the static theory was new at this point, and Einstein was not at all confident that there would be a positive result. “If no such deflection exists then the assumptions of the theory are not correct. One must keep in view precisely that these assumptions, as they already suggest themselves, are indeed really bold,” he wrote to Freundlich in September. “If only we had a much larger planet than Jupiter! But nature has not seen fit to concern itself with making it comfortable for us to figure out its laws.”43 The only feasible test would require a solar eclipse, and it would not necessarily be easy. Total solar eclipses are not that common, and the test would need one that reached totality over a region where observing conditions would be favorable and where instruments and photographic plates could be on hand to measure a very slight deflection. Also, it couldn’t be cloudy.44
Einstein continued to encourage Freundlich’s interest in conducting measurements to test the theory i
n early 1912: “It concerns a question of entirely fundamental significance. From the theoretical standpoint there seems to be a real probability that the effect actually exists.”45 Others were skeptical. Max von Laue, one of the earliest supporters of special relativity, fretted to Einstein: “Whether the astronomical test suggested by you is easily implemented, I don’t know; I fear however, that if the deflection is observed, it could always be attributed to the variation of the index of refraction of the solar atmosphere, over the composition of which there are also still many possible hypotheses.”46 Nevertheless, Freundlich seemed determined, and so Einstein wrote to the Viennese philosopher and physicist Ernst Mach—former doyen of Prague physics—in 1913 from his post at the ETH in Zurich that there would be an eclipse test of the new gravitational theory the following year.47
The plan was to go to the Crimea. Freundlich laid all the arguments out in a 1914 article and then headed in the summer to the Black Sea peninsula to set up his equipment.48 With eclipses, it’s all in the timing, and the timing was bad for a German astronomer to be traipsing around a strategically important peninsula in the Russian Empire. Freundlich and his team were interned when World War I broke out, and they were thus unable to make the measurements of the star field. (It turned out to be cloudy anyway, so even had they been at liberty it would not have made a difference.) The glory was left for the future, and for Eddington. Still, Einstein never forgot his earliest astronomical supporter, and in his preface to the third edition of Freundlich’s 1920 book on general relativity—the first to be published after the spectacular news of confirmation coming out of Britain—Einstein tipped his hat to Freundlich’s entrepreneurial intuition: “Mr. Freundlich is not only a renowned presenter of the matter as an expert of the present area of knowledge; he was also the first of his colleagues who had earnestly occupied himself with the testing of the theory.”49
Einstein in Bohemia Page 8